Chapter 3 — Motion In A Straight Line

Fig 1 — Educational Diagram: Introduction

Motion is the change in position of an object with time relative to a chosen reference frame. To describe motion in one dimension (a straight line) we use position (x), displacement (Δx), distance, velocity, speed and acceleration.

Reference frame and origin: Motion is described with respect to a coordinate system and a clock. Choose an origin and a positive direction on the straight line; the position x(t) gives the location at time t.

Distance vs Displacement: Distance is the total path length travelled (scalar, always >= 0). Displacement is a vector: Δx = x_final − x_initial; it can be positive, negative or zero.

Speed vs Velocity: Speed is a scalar measure of how fast an object moves. Velocity is a vector: it gives rate of change of displacement. Key distinctions:

• Average speed = total distance / total time.
• Average velocity = displacement / elapsed time = Δx/Δt.
• Instantaneous velocity v(t) = lim_{Δt→0} Δx/Δt = dx/dt.

Uniform and Non-uniform motion: If velocity is constant (v independent of t) motion is uniform — position changes linearly with time. If v varies with time, motion is non-uniform. When velocity changes, we define acceleration:

• Average acceleration = change in velocity / time interval = Δv/Δt.
• Instantaneous acceleration a(t) = dv/dt.

One-dimensional kinematic equations (for constant acceleration) give simple relations among displacement, velocity, acceleration and time. These are introduced here as the natural next step in studying motion in a straight line.

Why vectors matter: Even in one dimension, sign indicates direction. Positive/negative values of displacement, velocity or acceleration indicate which way along the chosen axis the object moves or accelerates.

Summary: The introductory ideas set up how to describe motion quantitatively — by specifying a reference frame, distinguishing scalar and vector quantities, defining average and instantaneous rates, and classifying motion as uniform or accelerated. Graphical representations of x–t, v–t and a–t are powerful tools for visualising motion.

Fig 2.1 — Educational Diagram: Projectile Motion Trajectory & Kinematic Equations

Definition: Physical quantities that can be described by a single number (with appropriate units) and no direction are called scalars. Quantities that require both magnitude and direction are called vectors.

Scalars: Examples include mass, temperature, time, speed, energy. Scalars add algebraically and are described by magnitude only.

Vectors: Examples include displacement, velocity, acceleration, force. A vector is represented graphically by an arrow: length corresponds to magnitude, arrowhead gives direction. In notation a vector is often written as \u03b1 or \vec{A}.

Properties and rules of vector algebra:

• Equality: Two vectors are equal if they have the same magnitude and direction, regardless of their initial points.
• Addition: Use the triangle rule (head-to-tail) or parallelogram rule to find the resultant R = A + B. Vector addition is commutative and associative.
• Subtraction: A - B = A + (−B), where −B is vector B reversed in direction.
• Multiplication by a scalar: Changes magnitude and possibly direction (if scalar is negative): kA has magnitude |k||A| and same or opposite direction depending on the sign of k.
• Zero vector: A vector of zero magnitude has no definite direction and acts as the additive identity.

Resolution into components: A vector in a plane can be resolved into orthogonal components. For a vector r making angle \theta with the x-axis, r = r_x î + r_y ĵ, where r_x = r cos \theta and r_y = r sin \theta. In coordinates, for r = (x, y), magnitude |r| = sqrt(x² + y²) and direction tan \theta = y/x.

One-dimensional motion: On a straight line vectors reduce to signed scalars. Displacement (vector) can be positive or negative depending on direction; distance (scalar) is always non-negative. Velocity is a vector (has sign), speed is scalar (non-negative).

Why vectors matter: Many physical laws involve directions (e.g., forces, velocities). Vector methods let you combine and resolve these quantities correctly. Some scalar results (like work) come from vector operations (work = force · displacement).

Fig 3 — Educational Diagram: Position and Displacement

Position: Position of a particle moving in a straight line is the location of the particle relative to a chosen origin and coordinate axis. In one dimension we usually denote the position by x (or s). The position is a vector quantity in general (position vector r), but in 1D it is represented by a signed number that gives both magnitude and direction from the origin. Unit: metre (m).

Displacement: Displacement is the change in position of the particle. If the initial position is x_i and the final position is x_f, the displacement Δx is defined as

Δx = x_f − x_i

Displacement is a vector (it has magnitude and direction). In 1D the sign of Δx indicates direction (e.g., positive to the right, negative to the left). Displacement depends only on the initial and final positions (it is path independent), and its magnitude can be less than or equal to the distance traveled.

Key points:

• Choose an origin and a positive direction before giving position or displacement.
• Distance is a scalar (total path length) and is always ≥ 0; displacement is a vector and can be zero, positive, or negative.
• If a particle returns to its starting point, its displacement is zero though distance may be nonzero.
• Average velocity over a time interval Δt is given by v_avg = Δx / Δt (connects displacement to motion). Instantaneous velocity is v = dx/dt.

Simple one-dimensional position vector notation: r = x âx (where âx is the unit vector along the chosen axis). Then Δr = r_f − r_i.

Fig 4 — Educational Diagram: Distance and Speed

Distance vs Displacement

Distance is the total length of the path travelled by an object; it is a scalar (has magnitude only) and is always non‑negative. Displacement is the shortest straight‑line change in position from the initial point to the final point; it is a vector (has magnitude and direction) and can be positive, negative or zero.

Speed vs Velocity

Speed is the rate of change of distance with time; it is a scalar. Velocity is the rate of change of displacement with time; it is a vector. Thus:

• Average speed = (total distance travelled) / (total time taken).
• Average velocity = (displacement) / (total time taken).
• Instantaneous velocity v(t) = dx/dt (derivative of displacement x with respect to time). Instantaneous speed is |v(t)| (magnitude of instantaneous velocity).

Uniform and Non‑uniform Motion

Uniform motion: speed (and velocity, if direction constant) are constant. On a displacement–time graph this appears as a straight line with constant slope. Non‑uniform motion: speed or velocity changes with time; the displacement–time graph is curved and the instantaneous velocity is given by the slope of the tangent to the curve.

Units and conversion

SI unit: metres per second (m/s). Common unit: kilometres per hour (km/h). Conversion: 1 m/s = 3.6 km/h, so v(km/h) = 3.6 × v(m/s).

Graphical interpretation

• Displacement–time graph: slope = velocity. A positive slope means motion in the positive direction; negative slope means motion in the opposite direction; zero slope means rest.
• Distance–time graph: always non‑decreasing; slope = speed (non‑negative). Flat parts mean rest; steeper parts mean higher speed.
• Velocity–time graph: the area under the curve between two times = displacement in that time interval. The sign of velocity indicates direction. For speed–time graph the area gives total distance travelled (if speed is used, area is always positive).

Important remarks

• Average speed = average of speeds only when the object spends equal time at each speed — in general average speed ≠ arithmetic mean of speeds; use total distance / total time.
• In circular motion at constant speed, speed is constant but velocity is not (direction keeps changing).

Fig 5 — Educational Diagram: Velocity

Definition: Velocity is a vector quantity that describes the rate of change of an object's displacement with respect to time. It gives both how fast and in which direction an object is moving.

Average velocity over a time interval Δt is defined as the displacement Δs divided by the time interval: v_avg = Δs/Δt. It depends on the net displacement, not on the path length.

Instantaneous velocity at time t is the derivative of displacement with respect to time: v(t) = ds/dt. Graphically, instantaneous velocity is the slope (gradient) of the displacement–time curve at that instant (the tangent slope).

Key properties:

• Vector quantity: has magnitude and direction. The sign of velocity indicates direction according to the chosen sign convention.
• Magnitude of velocity (|v|) is not always equal to speed; speed is the scalar (nonnegative) rate of motion while velocity can be negative.
• Units: SI unit is metres per second (m/s). Conversion: 1 m/s = 3.6 km/h.

Uniform vs non-uniform velocity: If velocity is constant (both magnitude and direction), motion is called uniform — displacement–time graph is a straight line and velocity–time graph is a horizontal line. If velocity changes, motion is non-uniform and the s–t curve is curved; instantaneous velocity is found from the tangent.

Relation with displacement and acceleration: v(t) = ds/dt, and displacement between times t1 and t2 is s(t2)−s(t1) = ∫_{t1}^{t2} v(t) dt (the area under a v–t graph). For motion with constant acceleration a (covered in later sections) v = u + at, where u is initial velocity.

Fig 6 — Educational Diagram: Acceleration

Definition: Acceleration is the rate of change of velocity with time. It is a vector quantity; in one-dimensional motion it has sign (positive or negative) depending on the chosen direction.

Mathematical forms:

• Average acceleration: a_avg = Δv / Δt = (v_final − v_initial) / (t_final − t_initial).
• Instantaneous acceleration: a = dv/dt = d²x/dt² (the derivative of velocity or the second derivative of position with respect to time).

Units and dimensions: SI unit is metre per second squared (m/s²). Dimension: [L T⁻²].

Signs and interpretation: If acceleration and velocity have the same sign, the speed increases. If they have opposite signs, the speed decreases (often called deceleration). Zero acceleration means constant velocity (including rest).

Uniform (constant) acceleration: When acceleration a is constant, the velocity and position satisfy the kinematic equations (for motion in a straight line):

• v = u + a t
• s = u t + (1/2) a t²
• v² = u² + 2 a s
• s = (u + v) t / 2 (average velocity × time)

Here u = initial velocity, v = velocity after time t, s = displacement in time t.

Non-uniform acceleration: When acceleration varies with time, a(t) = dv/dt. For a known a(t) you integrate to find v(t): v(t) = u + ∫ a(t) dt, and x(t) = x(0) + ∫ v(t) dt.

Graphical meaning: On a velocity–time (v–t) graph, acceleration equals the slope. On a position–time (x–t) graph acceleration is related to the curvature (concavity) — constant acceleration gives a parabolic x–t curve. On an acceleration–time (a–t) graph, the area under the curve between two times gives the change in velocity (Δv).

Special case — free fall: Near Earth’s surface, objects in free fall (neglecting air resistance) have approximately constant downward acceleration g ≈ 9.8 m/s². For upward motion under gravity, acceleration is −g (if upward is positive).

Key points to remember:

• Acceleration is about change of velocity (magnitude and/or direction). In straight-line motion direction is fixed, so acceleration changes the speed.
• Average vs instantaneous: average uses finite Δt, instantaneous uses derivative.
• Use signs consistently with your chosen positive direction.

Fig 7.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Definition: Motion in a straight line is the change of position (displacement) of a object along a straight path with time. It is classified into uniform motion and non-uniform motion depending on whether the velocity remains constant or not.

Uniform motion: An object executes uniform motion in a straight line if it covers equal displacements in equal intervals of time. In uniform motion the velocity (both magnitude and direction) is constant and acceleration is zero. Displacement is directly proportional to time.

• Mathematical statement: x(t) = x_0 + v t (where v = constant)
• Velocity: v = constant, a = 0
• Average velocity = instantaneous velocity = v

Non-uniform motion: Motion is non-uniform when an object covers unequal displacements in equal time intervals. Here velocity changes with time (in magnitude and/or direction). Acceleration is non-zero in general. For non-uniform motion, instantaneous quantities (dx/dt, dv/dt) are used.

• General relations: instantaneous velocity v(t) = dx/dt, instantaneous acceleration a(t) = dv/dt = d^2x/dt^2
• Displacement between t1 and t2: Δx = ∫_{t1}^{t2} v(t) dt

Special case — uniform (constant) acceleration: This is a simple and important type of non-uniform motion in which acceleration is constant (a = constant ≠ 0). The position and velocity then follow well-known kinematic formulas (given below).

How to tell from graphs: On an x–t (position–time) graph, uniform motion is a straight line; slope = velocity. Non-uniform motion gives a curved x–t graph. On a v–t graph, uniform motion is a horizontal line; changing v(t) indicates non-uniform motion. The slope of v–t is acceleration and the area under v–t gives displacement.

Units: displacement x in metres (m), velocity v in metres per second (m/s), acceleration a in metres per second squared (m/s^2).

Fig 8.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

Overview: When acceleration a of a particle moving in a straight line is constant (does not change with time), motion is called uniformly accelerated motion. Let u be the initial velocity (at t = 0), v the velocity at time t, s the displacement in time t, and a the constant acceleration. All quantities are one-dimensional (signs indicate direction).

Derivations (simple and calculus):

• Definition of acceleration: a = (v - u)/t. Rearranging gives the first equation: v = u + a t.
• For constant acceleration, average velocity = (u + v)/2. Displacement s in time t is average velocity × time, so s = ((u + v)/2) t. Substituting v = u + at yields the second standard form: s = u t + (1/2) a t².
• Eliminate t between v = u + at and s = u t + (1/2) a t² (or use v dv/ds = a): multiply v = u + at by v - u = at and rearrange to get v² = u² + 2 a s. This form is useful when time is not known.

Notes and conventions:

• Units: velocity in m/s, acceleration in m/s², displacement in m, time in s.
• Sign convention: choose a positive direction. If acceleration and initial velocity are opposite, a is negative (deceleration).
• Useful alternate form from s = ((u+v)/2) t: s = v t - (1/2) a t² by replacing u = v - at.
• These equations apply only when acceleration is constant.

Fig 9.1 — Educational Diagram: Newton's Laws of Motion & Free Body Diagrams

What it is
Motion under gravity refers to motion of objects when the only force (idealized) acting on them is gravity. Free fall is the special case when an object moves under gravity alone (air resistance neglected). The acceleration due to gravity near Earth's surface is constant in magnitude and denoted by g (approx. 9.8 m/s2).

Assumptions
1) Air resistance is negligible. 2) g is constant and directed vertically downward. 3) Upward and downward directions are chosen by a sign convention (take upward positive or downward positive consistently).

Key ideas and sign convention
If upward is taken positive, acceleration a = -g (vector points downward). If downward is positive, a = +g. The same kinematic equations for constant acceleration apply, with a = ±g.

Equations of motion applied to vertical motion
For an object with initial velocity u (vertical), after time t its velocity v and displacement s (vertical) satisfy the constant-acceleration relations:

• v = u + a t
• s = u t + 1/2 a t2
• v2 = u2 + 2 a s

Replace a by -g (if upward positive) or +g (if downward positive) to analyze free fall or vertical throws.

Common special cases
- Object dropped (u = 0): s = 1/2 g t2, v = g t, v2 = 2 g s.
- Object thrown upward (u upward): it decelerates at g, reaches maximum height when v = 0. Time to top t_top = u/g. Maximum height H = u2/(2 g). Total time to return to launch level (ignoring air resistance) = 2 u/g.
- Object thrown downward: initial velocity and gravity both increase speed; treat downward as positive or use a = -g with sign-correct u.

Independence of mass
In the absence of air resistance, all objects fall with the same acceleration g regardless of mass. (Galileo's result.)

Limitations
Real objects experience air resistance, leading to terminal velocity where drag equals weight. That requires more advanced treatment and is outside the idealized free-fall formulas.

Tips for solving problems
1) Choose sign convention and stick with it. 2) Plug a = ±g into kinematic equations. 3) Use v2 = u2 + 2 a s to eliminate time when needed. 4) Check units and physical reasonableness (e.g., time positive, heights nonnegative).

Fig 10 — Educational Diagram: Graphical Representation and Interpretation

Overview: Graphical representation is a way to describe motion in a straight line using plots of position (x), velocity (v) and acceleration (a) versus time (t) or position. Graphs provide immediate qualitative and quantitative information: slopes give rates (derivatives) and areas under curves give integrals (displacements or changes in velocity).

Key ideas:

• Instantaneous velocity is the slope (gradient) of the x–t curve: v(t) = dx/dt. A steeper slope means larger speed; slope sign gives direction.
• Instantaneous acceleration is the slope of the v–t curve: a(t) = dv/dt, and also the second derivative of position: a = d^2x/dt^2.
• Average values: average velocity over interval Δt is Δx/Δt (slope of the secant on x–t); average acceleration is Δv/Δt.
• Area meanings: area under v–t between t1 and t2 equals displacement Δx = ∫_{t1}^{t2} v dt. Area under a–t between t1 and t2 equals change in velocity Δv = ∫_{t1}^{t2} a dt.
• Common motion types: constant velocity → straight line on x–t; rest → horizontal x–t; uniform acceleration → parabola on x–t and straight line on v–t.

How to interpret graphs:

• From an x–t graph: slope at any point = instantaneous velocity; curvature indicates acceleration (concave up → positive acceleration, concave down → negative acceleration).
• From a v–t graph: slope = acceleration; sign of v gives direction of motion; area under curve = displacement. If v crosses zero, the object changes direction.
• From an a–t graph: constant nonzero line → uniform acceleration; zero line → constant velocity; area under curve gives change in speed/velocity.

Worked interpretation steps (typical procedure): Given a plot, first note axes and units, then determine slopes and areas for required quantities, check signs for direction, and where needed compute average values using secant slopes or definite integrals.

Fig 11 — Educational Diagram: Calculus Approach to Kinematics

Overview
In one-dimensional motion (motion in a straight line) calculus provides precise relationships between displacement, velocity and acceleration by using derivatives and integrals. Displacement s(t) is a function of time; velocity and acceleration are its first and second time-derivatives respectively.

Key definitions

• Displacement: s(t) — position as a function of time (can be positive or negative along a chosen axis).
• Instantaneous velocity: v(t) = ds/dt. It is the slope of the s–t curve at time t.
• Instantaneous acceleration: a(t) = dv/dt = d^2s/dt^2. It is the slope of the v–t curve at time t.

Average vs instantaneous
Average velocity over interval Δt: v_avg = (s(t+Δt) − s(t))/Δt. As Δt → 0 this becomes the instantaneous velocity v(t). Similarly for acceleration: a_avg = (v(t+Δt) − v(t))/Δt and a(t) = lim_{Δt→0} a_avg.

Integrals: reconstructing motion
Given acceleration a(t) and initial conditions v(t0)=v0, s(t0)=s0, you obtain velocity and position by integration: v(t) = v0 + ∫_{t0}^{t} a(τ) dτ and s(t) = s0 + ∫_{t0}^{t} v(τ) dτ. Thus area under a–t curve gives change in velocity; area under v–t curve gives displacement.

Useful derivative relation (chain rule)
When acceleration is expressed as a function of position, use a = dv/dt = (dv/ds)(ds/dt) = v (dv/ds). This often yields v dv = a ds which integrates to relations between v and s.

Constant acceleration (uniform a)
For constant acceleration a, integrals give the standard kinematic equations (with s0 and v0 at t = 0):

• s(t) = s0 + v0 t + 1/2 a t^2
• v(t) = v0 + a t
• v^2 = v0^2 + 2 a (s − s0) (obtained from integrating v dv = a ds)

Physical interpretation (graphs)
- On an s–t graph the tangent slope = instantaneous velocity; curvature indicates acceleration.
- On a v–t graph the slope = acceleration and the area under the curve = displacement.
- On an a–t graph the area = change in velocity.

Other concepts
- Jerk: j(t) = da/dt = d^3 s/dt^3, rate of change of acceleration (relevant in vehicle comfort, robotics).
- Sign convention: choose a positive direction; negative velocity or acceleration simply means motion or acceleration opposite that direction.

How to solve problems (steps)
1) Define coordinate axis and sign convention. 2) Write a(t) or v(t) if given. 3) Integrate (or differentiate) using initial conditions. 4) Use areas under curves or slopes of graphs where appropriate. 5) For variable a(s), use a = v dv/ds to relate v and s.

Summary: Calculus converts geometric ideas (slopes and areas) into precise relationships between position, velocity and acceleration enabling solution of a broad class of motion problems including variable acceleration.

Fig 12 — Educational Diagram: Relative Velocity in One Dimension

What is relative velocity?
Relative velocity describes how fast and in what direction one object appears to move from the reference frame of another object. In one dimension (motion along a straight line) it is simply the difference of their velocities measured in the same inertial frame.

Derivation (1D)
Let x_A(t) and x_B(t) be the positions of objects A and B measured in a common reference frame. The relative position of A with respect to B is x_{A/B}(t)=x_A(t)-x_B(t). Differentiate with respect to time:

v_{A/B} = \frac{d}{dt}x_{A/B}(t) = \frac{d}{dt}x_A(t) - \frac{d}{dt}x_B(t) = v_A - v_B.

This simple result means: the velocity of A relative to B equals the velocity of A in the chosen frame minus the velocity of B in the same frame. If you only care about how fast they approach or separate, use the relative speed: |v_{A/B}| = |v_A - v_B|.

Key points and interpretation

• Sign: v_{A/B} > 0 means A moves in the positive direction relative to B; v_{A/B} < 0 means it moves in the negative direction relative to B.
• Velocities must be expressed in the same coordinate system (same origin and axis direction).
• Relative acceleration follows the same rule: a_{A/B} = a_A - a_B.
• Galilean transformation: velocity measured in a frame moving at v_0 is v' = v - v_0. This is the same subtraction concept.

Useful application — meeting/overtaking
If two objects with constant velocities v_A and v_B start from positions x_{A0} and x_{B0}, they meet when x_A(t)=x_B(t). For constant velocities this gives t = (x_{B0} - x_{A0})/(v_A - v_B) provided v_A \neq v_B. The denominator is the relative velocity v_{A/B}.

Fig 13 — Educational Diagram: Solving Kinematics Problems: Strategies and Common Pitfalls

Solving kinematics problems in one dimension requires clear definitions, careful use of equations, and attention to signs and units. Kinematics describes how position, velocity and acceleration change with time without asking why they change (forces are not required). The basic tools are the definitions v = ds/dt and a = dv/dt, and for constant acceleration the standard SUVAT relations.

Step-by-step strategy

1. Read and visualize: Draw a simple diagram showing initial and final positions, direction of motion, and any reference points.
2. Choose sign convention: Pick a positive direction and stick to it for all quantities.
3. List knowns and unknowns: Write values for s (displacement), u (initial velocity), v (final velocity), a (acceleration), and t (time). Include units.
4. Decide if acceleration is constant: If constant, use SUVAT equations. If not, use calculus relations v = ds/dt and a = dv/dt and integrate/ differentiate as needed.
5. Select equations: Match the unknown you want to find with the equation that contains only knowns and that unknown (avoid adding extra unknowns).
6. Solve algebraically, then substitute numbers: Keep algebra symbolic as long as possible to reduce arithmetic errors.
7. Check units and signs: Ensure units are consistent (SI preferred) and sign of answers matches chosen direction.
8. Sanity-check results: Check special cases (t→0, a→0) and whether answer magnitude is reasonable.

Common pitfalls

• Confusing distance and displacement: Displacement is vector (can be negative); distance is scalar (always positive). Formulas use displacement.
• Incorrect sign convention: Forgetting that acceleration opposite to velocity is negative (deceleration) leads to wrong answers.
• Mixing average and instantaneous velocity: Average velocity over an interval is total displacement divided by total time. Instantaneous velocity is v(t) = ds/dt.
• Using a constant-acceleration formula when a varies: SUVAT equations only hold for uniform acceleration.
• Interpreting graphs wrongly: Slope of x–t gives velocity; slope of v–t gives acceleration; area under v–t gives displacement; area under a–t gives change in velocity.
• Unit mistakes: Mixing km/h with m/s without conversion causes large errors.
• Choosing the wrong equation: Pick the equation that does not introduce extra unknowns—e.g., use v^2 = u^2 + 2as when t is unknown.

Problem-solving tips

• Always state the frame of reference (origin and + direction).
• Label intermediate results and keep track of signs.
• For motion with piecewise constant acceleration, treat each segment separately and use continuity conditions at segment boundaries (position and velocity continuous).
• For variable acceleration given as a(t) or v(t), integrate: v(t)=v0+∫a(t)dt, s(t)=s0+∫v(t)dt.
• Use graphical methods where helpful: areas under v–t for displacement, slopes for instantaneous rates.

How to check your answer quickly

• Dimension check: each term in an equation must have same units.
• Limiting cases: if a→0, equations should reduce to constant-velocity results; if t→0, displacement should go to 0.
• Estimate: compare answer to rough estimate (e.g., distance = average speed × time).

Fig 14 — Educational Diagram: Special Topics and Extensions

Overview: "Special Topics and Extensions" collects useful extensions of one‑dimensional kinematics that go beyond constant acceleration: instantaneous (calculus) relations, motion with variable acceleration, motion under gravity including air resistance, relative motion in one dimension, braking/stopping distances, and the concept of jerk. These extensions link kinematics to simple dynamics and give tools for practical problems.

1. Instantaneous quantities and calculus relations
Instantaneous velocity and acceleration are defined as derivatives: v(t)=dx/dt and a(t)=dv/dt. These give displacement and velocity by integration: x(t)=x(t0)+∫_{t0}^{t}v(t')dt' and v(t)=v(t0)+∫_{t0}^{t}a(t')dt'.

2. Motion with variable acceleration
When a is not constant you can use a(t)=dv/dt or the useful chain rule form a = v dv/dx (so v dv = a dx). Integrate the appropriate relation to find v(t) or v(x) and then x(t).

3. Motion under gravity (vertical motion)
Near Earth’s surface use constant acceleration g (≈9.8 m/s² downwards). For motion with upward initial speed u: v = u - gt, y = y0 + ut - 1/2 g t², v² = u² - 2g(y - y0). Use sign convention consistently.

4. Motion with resistive forces (terminal velocity)
- Linear drag: F_drag = -k v → m dv/dt = mg - k v (for downward motion). Solution: v(t) = v_T(1 - e^{-t/τ}), where v_T = mg/k and τ = m/k.
- Quadratic drag (high speed): F_drag ≈ -c v² → terminal speed v_T = sqrt(mg/c). These models explain approach to terminal velocity (skydiver, raindrops) and require solving ODEs (separable or linear).

5. Relative motion in one dimension
Velocities add/subtract depending on direction. If A moves with v_A and B with v_B (measured in same inertial frame), velocity of A relative to B is v_{A/B} = v_A - v_B. Displacements and times transform accordingly for different inertial observers (Galilean transformation).

6. Braking, stopping distance and safety
Stopping distance = reaction distance + braking distance. Reaction distance = u t_reaction. Braking distance (constant deceleration a): d = u²/(2|a|). These follow from kinematics and are used in road‑safety calculations.

7. Jerk and smoothness
Jerk j = da/dt is rate of change of acceleration; important in engineering (elevator, vehicle comfort) though not central to introductory problems.

How to choose methods: For time‑known problems integrate a(t); when acceleration depends on x use v dv/dx; when time not required use v² - u² = 2aΔx (energy‑like relation). For resistive forces set up Newton’s 2nd law and solve ODE for v(t).